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12        XII   TOPOLOGY AND TOPOLOGICAL ALGEBRA

Two families of pseudo-distances ( da)a 6 T , (d^)^ e L on the same set E are
said to be topologically equivalent if they define the same topology on E.

(12.4.5) If (Wa)a6i is any family of pseudo-distances on a set E, there exists a
topologically equivalent family (d^)(xel such that 0 ^ d'n ^ 1 for all a e L

Let <p(u) = inf(w, 1) for 0 ^ u < + oo. Then <p(0) = 0, and (p is an increas-
ing function on [0, 4- oo[ and satisfies the inequality

(p(u + v) :g <p(w) + <p(tf).

(This last assertion is clear if u> 1 or i; > 1 or if w + i? ^ 1; and if u ^ 1 and
v < 1 and i/ + t; > 1 we have <p(w + u) = 1 < <p() + <p(f).) It follows now from
(12.4.1) that d'a = <p  da is a pseudo-distance on E, for each a e I. That the
topologies defined by the families (da) and (d) are the same follows from
(12.2.1) and the fact that, when 0<r< 1, the relations dx(x,y)<r and
^(X JO < r are equivalent.

(12.4.6) A Hausdorff uniformizable space E, w/zo^ topology can be defined by
a sequence (dn) of pseudo-distances, is metrizable.

Without loss of generality we may assume that the sequence (dn) is infinite,
and by (12.4.5) that 0 ^ dn ^ 1 for all n. Then the series

(1 2.4.6.1 )    d(x, y)  1 rf^jc, y) + ~ d2(x, y) + ~- +  dn(x, y) + --

converges for all pairs (x, y) of elements of E, and d is a pseudo-distance on
E (12.4.1). Moreover, d is a distance, because d(x, j>) = 0 implies that
dn(x9 7) = 0 fr a^ ^> anc^ therefore that x = y because E is Hausdorff (12.4.4).
If B0(:x;; r) is the open ball with center x and radius r with respect to the dis-
tance d, then it follows immediately from (1 2.4.6.1 ) that BO(JC ; r) c E(x ; , 2V)
for all n. Conversely, if n is so large that 2n~l ^ 1/r, then we have

for all x, 7 in E, and therefore

B(*; (1,2,..., n), (ir, . . . , ir)) cz BO(JC, r).
By (12.2.1), the proof is complete.

(12.4.7) Let E be a topological space and (Un) a finite or denumerable open
covering ofE such that the subspaces Uw are separable and metrizable. Then E is
separable and metrizable.? ^/2) of x and 7, respectively, are disjoint, by virtue of the triangle